(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 2044231, 38868] NotebookOptionsPosition[ 2026823, 38336] NotebookOutlinePosition[ 2027234, 38354] CellTagsIndexPosition[ 2027191, 38351] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"2", "-", "2"}]], "Input", CellChangeTimes->{{3.417090267567175*^9, 3.417090267793248*^9}}], Cell[BoxData["0"], "Output", CellChangeTimes->{3.417090270374423*^9, 3.4172078172204227`*^9, 3.417273195038809*^9, 3.568663710785192*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"CompConj", "[", "A_", "]"}], " ", ":=", " ", RowBox[{"A", " ", "/.", "\[InvisibleSpace]", RowBox[{ RowBox[{"Complex", "[", RowBox[{"0", ",", "n_"}], "]"}], "->", RowBox[{"-", RowBox[{"Complex", "[", RowBox[{"0", ",", "n"}], "]"}]}]}]}]}]], "Input"], Cell[CellGroupData[{ Cell["Quantum Lissajous Motion", "Title", CellChangeTimes->{{3.417090294026064*^9, 3.417090303461097*^9}}], Cell["\<\ Nicholas Wheeler ÌÇÐÄÊÓÆµ Physics Motion 13 April 2008\ \>", "Text", CellChangeTimes->{{3.4170903117065697`*^9, 3.417090338695719*^9}}, FontSize->10], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.417090387952009*^9}], Cell[CellGroupData[{ Cell["Introduction", "Subsection", CellChangeTimes->{{3.417090393060527*^9, 3.417090396092708*^9}}], Cell["\<\ A few days ago I received a request from the editor of the American Journal \ of Physics to referee a manuscript submitted by Meredith A. Mahoney, David M. \ Paganin & Helen M. L. Faulkner, all of whom are/were attached to the School \ of Physics, Monash University in Victoria, Australia. Their paper bears the \ title \"Quantum-mechanical phase vortices in the two-dimensional infinite \ square well.\" The School of Physics maintains an elaborate web site, which, \ however, makes no mention of either Mahoney or Paganin, who I suspect were \ students of Faulkner\ \>", "Text", CellChangeTimes->{{3.417090412926281*^9, 3.417090461205743*^9}, { 3.4170904946239233`*^9, 3.417090585297188*^9}, {3.4170906718202953`*^9, 3.4170906857147713`*^9}, {3.417090720475644*^9, 3.417090769613051*^9}, { 3.4170918682050543`*^9, 3.417091874382893*^9}, {3.417091911847557*^9, 3.417092066298967*^9}, {3.417092213714038*^9, 3.417092272628023*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Hyperlink", "[", RowBox[{ "\"\\"", ",", "\"\\""}], "]"}]], "Input", CellChangeTimes->{{3.417091178587617*^9, 3.417091201075075*^9}}], Cell[BoxData[ TagBox[ ButtonBox[ PaneSelectorBox[{False->"\<\"Faulkner\"\>", True-> StyleBox["\<\"Faulkner\"\>", "HyperlinkActive"]}, Dynamic[ CurrentValue["MouseOver"]], BaseStyle->{"Hyperlink"}, BaselinePosition->Baseline, FrameMargins->0, ImageSize->Automatic], BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.physics.monash.edu.au/staff/faulkner.html"], None}, ButtonNote->"http://www.physics.monash.edu.au/staff/faulkner.html"], Annotation[#, "http://www.physics.monash.edu.au/staff/faulkner.html", "Hyperlink"]& ]], "Output", CellChangeTimes->{3.4170912075244703`*^9, 3.568663713341762*^9}] }, Open ]], Cell[TextData[{ "Faulkner lists electron microscopy, diffraction physics, phase retrieval \ and ", StyleBox["optical vortices", FontVariations->{"Underline"->True}], " as primary interests, and co-authored papers\[LongDash]one entitled \ \"Phase retrieval and aberration correction in the presence of vortices in \ high-resolution transmission electron microscopy\"\[LongDash]with Paganin in \ 2001 and 2003. I suspect she is the source of the references to \"phase \ vortices\" in the paper (emphasized in its title), which appear to me to be \ entirely extraneous to the paper's small but interesting point.\n\nHere I \ describe the path\[LongDash]the rectilinear Lissajous figure\[LongDash]traced \ by a particle that bounces around elastically in a square box. 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The motion is periodic, with period ", "InlineFormula"], StyleBox[" \[Tau] = 2a/v", "Output"], "\[NonBreakingSpace], where ", StyleBox["v", "Output"], " refers to its constant speed. The situation is too simple to discuss; my \ challenge here is to teach ", StyleBox["Mathematica", FontSlant->"Italic"], " ", StyleBox["how to draw", FontVariations->{"Underline"->True}], " such an ", StyleBox["x[t]", "Output"], "." }], "Text", CellChangeTimes->{{3.417093117867782*^9, 3.4170931688662157`*^9}, { 3.417093215300877*^9, 3.417093281581341*^9}, {3.4170933222980843`*^9, 3.417093390806628*^9}, {3.4170934321543093`*^9, 3.417093518038028*^9}, { 3.4170935751613207`*^9, 3.417093579700493*^9}}], Cell[TextData[{ "I will assume henceforth that the box is a ", StyleBox["unit box ", FontSlant->"Italic"], "(", StyleBox["a = 1", "Output"], "). I assume that the particle is launched from ", StyleBox["0 < x0 < 1 ", "Output"], "and that initially it moves to the right with speed ", StyleBox["v", "Output"], ". 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This turns out \ to be a subject with a literature \ \>", "Text", CellChangeTimes->{{3.4171346697467127`*^9, 3.417134773217217*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Hyperlink", "[", RowBox[{ "\"\\"", ",", " ", "\"\\""}], " ", "]"}]], "Input", CellChangeTimes->{{3.416950398809861*^9, 3.41695045548628*^9}, { 3.4169504982585382`*^9, 3.4169505006712103`*^9}, 3.4169506445243673`*^9}], Cell[BoxData[ TagBox[ ButtonBox[ PaneSelectorBox[{False->"\<\"Quantum mechanical Lissajous figures\"\>", True-> StyleBox["\<\"Quantum mechanical Lissajous figures\"\>", "HyperlinkActive"]}, Dynamic[ CurrentValue["MouseOver"]], BaseStyle->{"Hyperlink"}, BaselinePosition->Baseline, FrameMargins->0, ImageSize->Automatic], BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.google.com/search?hl=en&q=quantum+mechanical+Lissajous&\ btnG=Search"], None}, ButtonNote-> "http://www.google.com/search?hl=en&q=quantum+mechanical+Lissajous&btnG=\ Search"], Annotation[#, "http://www.google.com/search?hl=en&q=quantum+mechanical+Lissajous&btnG=\ Search", "Hyperlink"]& ]], "Output", CellChangeTimes->{3.417195799138186*^9, 3.417214803410429*^9, 3.568663735744883*^9}] }, Open ]], Cell["\<\ Much of this work has to do with semi-classical quantum mechanics (the area \ where classical and quantum mechanics meet), with is a subject that is still \ being quite actively pursued, and which has generated a vast literature with \ deep ramifications: see \ \>", "Text", CellChangeTimes->{{3.4171348434474363`*^9, 3.417134972309964*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Hyperlink", "[", RowBox[{ "\"\\"", ",", "\"\\""}], "]"}]], "Input", CellChangeTimes->{{3.416950831284782*^9, 3.416950886824874*^9}, { 3.417135000839862*^9, 3.417135009746335*^9}}], Cell[BoxData[ TagBox[ ButtonBox[ PaneSelectorBox[{False->"\<\"Large semi-classical bibliography\"\>", True-> StyleBox["\<\"Large semi-classical bibliography\"\>", "HyperlinkActive"]}, Dynamic[ CurrentValue["MouseOver"]], BaseStyle->{"Hyperlink"}, BaselinePosition->Baseline, FrameMargins->0, ImageSize->Automatic], BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.citebase.org/abstract?identifier=oai%3AarXiv.org%3Aquant-\ ph%2F0611062&action=citeshits&citeshits=cites"], None}, ButtonNote-> "http://www.citebase.org/abstract?identifier=oai%3AarXiv.org%3Aquant-ph%\ 2F0611062&action=citeshits&citeshits=cites"], Annotation[#, "http://www.citebase.org/abstract?identifier=oai%3AarXiv.org%3Aquant-ph%\ 2F0611062&action=citeshits&citeshits=cites", "Hyperlink"]& ]], "Output", CellChangeTimes->{3.41719584157824*^9, 3.417214809431924*^9, 3.568663735803866*^9}] }, Open ]], Cell["\<\ To expose the points at issue it is sufficient to examine an illustrative \ case. Use the material now in hand to construct the x,y-separated product \ state\ \>", "Text", CellChangeTimes->{{3.4171351013169117`*^9, 3.4171351525423813`*^9}, 3.417195881918285*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"\[CapitalPsi]", "[", RowBox[{ "x_", ",", " ", "y_", ",", " ", "t_", ",", " ", "\[Alpha]_", ",", " ", "\[Beta]_", ",", " ", "\[Delta]_"}], "]"}], " ", ":=", RowBox[{ RowBox[{"\[CapitalPsi]13", "[", RowBox[{"x", ",", "t", ",", "\[Alpha]"}], "]"}], RowBox[{"\[CapitalPsi]13", "[", RowBox[{"y", ",", RowBox[{"t", "+", "\[Delta]"}], ",", "\[Beta]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.417190942612501*^9, 3.4171909708196907`*^9}, { 3.417191966681527*^9, 3.417191969349207*^9}, {3.417192340982568*^9, 3.4171923413194532`*^9}}], Cell[BoxData["$Failed"], "Output", CellChangeTimes->{3.568663735915628*^9}] }, Open ]], Cell["\<\ Dispite the simplicity of that construction, the resulting expressions for \ probability density and probability current become much too lengthy to permit \ exploration of this subject with pen and ink on paper. Animated graphics \ provide the only comprehending the meaning of the results to which we will be \ led. Those two circumstances probably explain why the subject is not treated \ in any of the standard textbooks.\ \>", "Text", CellChangeTimes->{{3.417191104562354*^9, 3.417191213720109*^9}, { 3.417191257586162*^9, 3.417191308658143*^9}, {3.417214868425455*^9, 3.417214937753861*^9}, {3.417215002931637*^9, 3.417215011275384*^9}}], Cell["\<\ Before proceeding further, we verify that the function thus defined is indeed \ a solution of the Schr\[ODoubleDot]dinger equation:\ \>", "Text", CellChangeTimes->{{3.4172010882529488`*^9, 3.417201118821793*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", RowBox[{"D", "[", RowBox[{ RowBox[{"\[CapitalPsi]", "[", RowBox[{ "x", ",", " ", "y", ",", " ", "t", ",", " ", "\[Alpha]", ",", " ", "\[Beta]", ",", " ", "\[Delta]"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "2"}], "}"}]}], "]"}]}], "-", RowBox[{"D", "[", RowBox[{ RowBox[{"\[CapitalPsi]", "[", RowBox[{ "x", ",", " ", "y", ",", " ", "t", ",", " ", "\[Alpha]", ",", " ", 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